Transforming the Transformer

transforming the transformer

In the last post, Design of an all-pass Hilbert Transformer filter, we began with the pole/zero pattern shown in Figure 1:

Figure 1 - Hilbert Configuration in the s-plane

We then followed the process:

1. Divided the filter into 2 halves called filter A and B
2. Transformed into the z-plane
3. Added a unit delay in series with filter A

Which gave us a Hilbert Transformer - a filter bank that takes an input and produces 2 related outputs with phase difference 90°. You should read that post to remind yourself how the filter was divided into two - by interleaved poles and zeros (and not by the 2 lines shown on the s-plane)


There is an additional step in the above sequence that we can perform:

0. Transform in the s-plane

pole/zero translations

For example, figure 2 shows a shift or translation of -pi/2:

Figure 2 - shifting the Hilbert configuration by pi/2

This results in a 90° rotation in the z-plane compared with the Hilbert Transformer:

Figure 3 - both (shifted) filters plotted in the z-plane (distant zeros at  +/- 23.14i and +/- 4.81i omitted)

I have plotted both filters together but they should be considered as separate interleaved filters.

We can then follow steps 1,2 and 3 of the process to produce a pair of all-pass sections with the following phase response:

Figure 4 - phase difference between filter B and delayed filter A

This new filter bank produces a pair of outputs 180° apart in the range 0 Hz to pi/2 Hz and 0° apart in the range pi/2 - pi Hz.

If you sum the outputs of the filter, the low frequencies (0 < f < pi/2) cancel out whilst the high frequencies (pi/2 < f < pi) reinforce each other. The result is a high-pass filter with a cut-off frequency at pi/2 Hz

On the other hand, if you take the difference of the filter outputs the frequencies combine in the inverse sense and you have a low-pass filter. So we get 2 filters for the price of 1 - a bargain.

[We can equally do the pole/zero translation directly in the z-plane just by rotating the Hilbert configuration through 90° to take it from the real axis to the imaginary axis.]

transforming the transformation

There are additional geometric manipulations we can perform in the s-plane (or indeed the z-plane) to alter the transition frequency of the phase response of figure 4.

pole/zero scaling

We can break the pi separation of the two s-plane rows by squeezing them together (or moving them apart). 

Figure 5 - Hilbert shifted by pi/2 and squeezed by pi/2

This "pi/2 X pi/2" operation results in the z-plane distribution:

Figure 6 - the pair of interleaved (shifted and squeezed) filters mapped into the z-plane
(distant zeros at 16.36 +- 16.36i omitted for clarity)

Again, we follow steps 1,2 and 3 of the process to give the phase relationship:

Figure 7 - adjusted phase difference between filter B and delayed filter A

This new relationship has produced another low/high pass filter pair with the transition frequency now at pi/4.

You can easily see that by applying different degrees of 'squeeze' or 'stretch' on the pair of s-plane lines we can change the transition frequency to whatever we like. The only caveat is that when the transition frequency moves close to 0 or pi, you have to reduce the order of the filter (i.e. use a lower 'n' in the filter generator function) due to the poles and zeros of the z-plane conjugate pairs interfering with one another.

RAW~